| Résume | Abstract: Given a family of smooth projective varieties, one can
consider the relative de Rham moduli space, of flat vector bundles of
rank n on the fibers. The flat vector bundles which underlie a
Z-polarized variation of Hodge structure form the "non-abelian Hodge
locus". Simpson proved that this locus is closed and analytic, and he
conjectured it is algebraic. Simpson's conjecture would imply a
conjecture of Deligne that only finitely many representations of the
fundamental group underlie a Z-PVHS, on some fiber. I will discuss a
proof of Deligne’s and Simpson’s conjectures, under the additional
condition that the Z-Zariski closure of monodromy is cocompact. This is
joint work with Salim Tayou.
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